A Riesel number is a positive integer k
for which the integers
k.2n-1
are all composite (that is, for every
positive integer n). In 1956 Riesel formed a set
of congruences whose solutions k had this
property (so there is an infinite number of Riesel
numbers). In particular, Riesel showed that the
multiplier k=509203 had this property (as did
509203 plus any positive integer multiple of 11184810).

It is conjectured that k=509203
is the smallest Riesel number. To show that it is the
smallest, one needs to find a prime of the form
k.2n-1 for each of the positive integers k less than 509203. Primes
have already been found for most of these k's.
Wilfrid Keller is currently organizing a search to find
primes for the remaining values.

Notice that the Riesel numbers are very similar to
the Sierpinski numbers. The latter numbers were explored by Sierpinski in 1960, several years after
Riesel wrote his paper, and their infinitude was proven using very similar congruences to those Riesel used.
Ironically, the Sierpinski number received much more attention than the Riesel numbers, perhaps because the
Riesel paper was in Swedish.

P. Ribenboim, The new book of prime number records, 3rd edition, Springer-Verlag, New York, NY, 1995. pp. xxiv+541, ISBN 0-387-94457-5. MR 96k:11112[An excellent resource for those with some college mathematics. Basically a Guinness Book of World Records for primes with much of the relevant mathematics. The extensive bibliography is seventy-five pages.]